A More Efficient 1–Checkable Secure Outsourcing Algorithm for Bilinear Maps

  • Öznur KalkarEmail author
  • Mehmet Sabir Kiraz
  • İsa Sertkaya
  • Osmanbey Uzunkol
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10741)


With the rapid advancements in innovative technologies like cloud computing, internet of things, and mobile computing, the paradigm to delegate the heavy computational tasks from trusted and resource-constrained devices to potentially untrusted and more powerful services has gained a lot of attention. Ensuring the verifiability of the outsourced computation along with the security and privacy requirements is an active research area. Several cryptographic protocols have been proposed by using pairing-based cryptographic techniques based on bilinear maps of suitable elliptic curves. However, the computational overhead of bilinear maps forms the most expensive part of those protocols. In this paper, we propose a new 1–checkable algorithm under the one-malicious version of a two-untrusted-program model. Our solution is approximately twice as efficient as the single comparably efficient 1–checkable solution in the literature, and requires only 4 elliptic curve point additions in the preimage and 6 field multiplications in the image of the bilinear map.


Outsourcing computation Bilinear maps Security Privacy 


  1. 1.
    Arabacı, O., Kiraz, M.S., Sertkaya, I., Uzunkol, O.: More efficient secure outsourcing methods for bilinear maps. Cryptology ePrint Archive, Report 2015/960 (2015).
  2. 2.
    Barbulescu, R., Duquesne, S.: Updating key size estimations for pairings. Cryptology ePrint Archive, Report 2017/334 (2017).
  3. 3.
    Barreto, P., Galbraith, S., Higeartaigh, C., Scott, M.: Efficient pairing computation on supersingular abelian varieties. Des. Codes Cryptogr. 42(3), 239–271 (2007). Scholar
  4. 4.
    Beuchat, J.-L., González-Díaz, J.E., Mitsunari, S., Okamoto, E., Rodríguez-Henríquez, F., Teruya, T.: High-speed software implementation of the optimal ate pairing over Barreto–Naehrig curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 21–39. Springer, Heidelberg (2010). Scholar
  5. 5.
    Blake, I., Seroussi, G., Smart, N.: Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series. Cambridge University Press, New York (2005)CrossRefGoogle Scholar
  6. 6.
    Boneh, D., Franklin, M.: Identity-based encryption from the weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001). Scholar
  7. 7.
    Boneh, D., Lynn, B., Shacham, H.: Short signatures from the weil pairing. J. Cryptol. 17(4), 297–319 (2004). Scholar
  8. 8.
    Canard, S., Devigne, J., Sanders, O.: Delegating a pairing can be both secure and efficient. In: Boureanu, I., Owesarski, P., Vaudenay, S. (eds.) ACNS 2014. LNCS, vol. 8479, pp. 549–565. Springer, Cham (2014). Scholar
  9. 9.
    Chen, X.: Introduction to secure outsourcing computation. Synth. Lect. Inf. Secur. Priv. Trust 8(2), 1–93 (2016)Google Scholar
  10. 10.
    Chen, X., Susilo, W., Li, J., Wong, D., Ma, J., Tang, S., Tang, Q.: Efficient algorithms for secure outsourcing of bilinear pairings. Theor. Comput. Sci. 562, 112–121 (2015). Scholar
  11. 11.
    Chevallier-Mames, B., Coron, J.S., McCullagh, N., Naccache, D., Scott, M.: Secure delegation of elliptic-curve pairing. Cryptology ePrint Archive, Report 2005/150 (2005)Google Scholar
  12. 12.
    Galbraith, S.D., Paterson, K.G., Smart, N.P.: Pairings for cryptographers. Discrete Appl. Math. 156(16), 3113–3121 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hess, F., Smart, N., Vercauteren, F.: The eta pairing revisited. IEEE Trans. Inf. Theory 52(10), 4595–4602 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hohenberger, S., Lysyanskaya, A.: How to securely outsource cryptographic computations. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 264–282. Springer, Heidelberg (2005). Scholar
  15. 15.
    Joux, A.: A one round protocol for tripartite Diffie-Hellman. J. Cryptol. 17(4), 263–276 (2004). Scholar
  16. 16.
    Kang, B.G., Lee, M.S., Park, J.H.: Efficient delegation of pairing computation (2005)Google Scholar
  17. 17.
    Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005). Scholar
  18. 18.
    Kumar, K., Liu, J., Lu, Y.H., Bhargava, B.: A survey of computation offloading for mobile systems. Mob. Netw. Appl. 18(1), 129–140 (2013). Scholar
  19. 19.
    Lin, X.J., Qu, H., Zhang, X.: New efficient and flexible algorithms for secure outsourcing of bilinear pairings. Cryptology ePrint Archive, Report 2016/076 (2016).
  20. 20.
    Luo, Y., Fu, S., Huang, K., Wang, D., Xu, M.: Securely outsourcing of bilinear pairings with untrusted servers for cloud storage. In: Trustcom/BigDataSE/ISPA (2016)Google Scholar
  21. 21.
    Mell, P., Grance, T.: The NIST definition of cloud computing. NIST Special Publication, pp. 800–145 (2011)Google Scholar
  22. 22.
    Nguyen, P.Q., Shparlinski, I.E., Stern, J.: Distribution of modular sums and the security of the server aided exponentiation (2000)Google Scholar
  23. 23.
    Ren, Y., Ding, N., Wang, T., Lu, H., Gu, D.: New algorithms for verifiable outsourcing of bilinear pairings. Sci. China Inf. Sci. 59(9), 99103 (2016)CrossRefGoogle Scholar
  24. 24.
    Shacham, H.: New paradigms in signature schemes. Ph.D. thesis, Stanford, CA, USA (2006)Google Scholar
  25. 25.
    Tian, H., Zhang, F., Ren, K.: Secure bilinear pairing outsourcing made more efficient and flexible. In: Proceedings of the 10th ACM Symposium on Information, Computer and Communications Security, ASIA CCS 2015, pp. 417–426. ACM, New York (2015).
  26. 26.
    Wang, Y., Wu, Q., Wong, D.S., Qin, B., Chow, S.S.M., Liu, Z., Tan, X.: Securely outsourcing exponentiations with single untrusted program for cloud storage. In: Kutyłowski, M., Vaidya, J. (eds.) ESORICS 2014. LNCS, vol. 8712, pp. 326–343. Springer, Cham (2014). Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  • Öznur Kalkar
    • 1
    Email author
  • Mehmet Sabir Kiraz
    • 1
  • İsa Sertkaya
    • 1
  • Osmanbey Uzunkol
    • 2
  1. 1.Mathematical and Computational SciencesTÜBİTAK BİLGEMKocaeliTurkey
  2. 2.Faculty of Mathematics and Computer ScienceFernUniversität in HagenHagenGermany

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